From a couple quick U-boob viewings of actual cabs, I've come to the conclusion that it depends on the machine as to how the P1 & P2 lights flash...
-This 1 seems to flash alternatively (hard to tell for sure) https://youtu.be/FQA_NpZUC24?t=26s
while the rest of them all flashed in unison. https://youtu.be/OE7BZAq8Noo?t=47s
-But, on all of them the P1 button stayed on after hitting it to start the game. I couldn't see on any of them if it switched to the other button & glowed as the players changed or if both of them came on & stayed on. (but I seem to recall it did the former...)
So no definitive answer here.
Have you tried all the different versions? Do they all work the same now?
Hope this helps a bit...GL!
Amazing how something that seems so simple ends up being a such a PITA to figure out...
Pessimist: Oh, this can't get any worse!
Optimist: Yes, it can!